Quasi-classical distribution

A complete description of the method requires a procedure for selecting the initial conditions. At t 0, initial values for the complex basis set coefficients and the parameters that define the nuclear basis set (position, momentum, and nuclear phase) must be provided. Typically at the beginning of the simulation only one electronic state is populated, and the wavefunction on this state is modeled as a sum over discrete trajectories. The size of initial basis set (N/it = 0)) is clearly important, and this point will be discussed later. Once the initial basis set size is chosen, the parameters of each nuclear basis function must be chosen. In most of our calculations, these parameters were drawn randomly from the appropriate Wigner distribution [65], but the earliest work used a quasi-classical procedure [39,66,67], At this point, the complex amplitudes are determined by projection of the AIMS wavefunction on the target initial state (T 1)... 

The physical origin of the dispersion interaction is often described in terms of a quasi-classical induced-dipole-induced-dipole picture. The quantum-mechanical fluctuations of the electronic distribution about its spherically symmetric average can be pictured as leading to an instantaneous (snapshot) dipole /za(mst) on monomer a, which in turn induces an instantaneous dipole tb(mst) on b. Thus, if the dipole fluctuations of the two monomers are properly correlated, a net attraction of the form (5.25) results. As remarked by Hirschfelder et al,28... 

That is, the classical DoF propagate according to a mean-field potential, the value of which is weighted by the instantaneous populations of the different quantum states. A MFT calculation thus consists of the self-consistent solution of the time-dependent Schrodinger equation (28) for the quantum DoF and Newton s equation (32) for the classical DoF. To represent the initial state (15) of the molecular system, the electronic DoF dk Q) as well as the nuclear DoF xj Q) and Pj 0) are sampled from a quasi-classical phase-space distribution [23, 24, 26]. 

There is, however, an important conceptional difference between the two approaches. On the quasi-classical level, this difference simply manifests itself in the initial conditions chosen for the electronic DoF. Let us consider an electronic two-level system that is initially assumed to be in the electronic state vl/]). In the mean-field formulation, the initial conditions are action-angle variables [cf. Eq. (18)], the electronic initial distribution in (90), is given by pgj = 6 Ni — 1)6 N2). In the mapping formalism, on the other hand, the initial electronic state vl/]) is represented by the first oscillator being in its first excited state and second oscillator being in its ground state [cf. Eq. (91)]. This corresponds to the... 

Figure 3.28. N2 vibrational state distribution in associative desorption from Ru(0001). (a) Observed in experiment. From Ref. [126]. (b) From 3D (Z, R, q) first principles quasi-classical dynamics, with the solid triangles pointing upward being adiabatic dynamics and the squares from molecular dynamics with electronic frictions also from DFT. Based on the PES and frictions of Ref. [68]. The open triangles pointing downward are the results of 6D first principles adiabatic quasi-classical dynamics from Ref. [253].

Halberstadt, N., Beswick, J.A., and Schinke, R. (1991). Rotational distributions in the vibrational predissociation of weakly bound complexes Quasi-classical golden rule treatment, in Half Collision Resonance Phenomena in Molecules Experimental and Theoretical Approaches, ed. M. Garcia-Sucre, G. Raseev, and S.C. Ross (American Institute of Physics, New York). 

Figure 5 Quasi-classical probability distribution for the asymptotic total product HD energy, for H incident on D-covered Cu(l 11), for an incident H atom energy of 70 meV.

Figure 7 Product HD rotational state distributions, from the experiments of Ref. [16] (open squares) and from the quasi-classical calculations (filled squares).

The accuracy of trajectory calculations have been examined by comparing the results of exact quantum and quasi-classical calculations [143], The most difficult problem lies in the selection of a method for quantising the continuous classical product energy distributions. There is no formal justification for such a procedure, but it enables comparison with experimental vibrational and rotational distributions. No single method appears to be suitable for all systems. 

Carpenter, B. K. Bimodal distribution of lifetimes for an intermediate from a quasi-classical dynamics simulation, J. Am. Chem. Soc. 1996,118, 10329-10330. 

In the full-quantum dynamics method, the distribution of nuclear positions is accounted for in nuclear wavepacket form, that is, by a function that defines the distribution of momenta of each atom and the distribution of the position in the space of each atom. In classical and semi-classical or quasi-classical dynamics methods, the wavepacket distribution is emulated by a swarm of trajectories. We now briefly discuss how sampling can generate this swarm. 

Figure 2.31 Distribution of the energy at the hop geometry of the model chromophore Im (computed by combined quasi-classical CASSCF/3-21G trajectory calculations) for the three series of calculations presented in Figure 2.30.

Figure 2.32 Distribution (for 100 CASSCF/3-21G combined quasi-classical trajectories) of the lifetimes (r) and torsion angles (ff) of the hop geometry of model chromophore Im.

To simulate collisions between a gaseous molecule and a surface, Cartesian coordinates and momenta are first chosen from the energies for the molecule and surface following procedures described above. The distribution of energies for a surface at temperature Ts may be chosen by either classical or quasi-classical normal-mode sampling. From classical statistical mechanics... 

In the case of coherent laser light, the pulses are characterized by well-defined phase relationships and slowly varying amplitudes (Haken, 1970). Such quasi-classical light pulses have spectral and temporal distributions that are also strictly related by a Fourier transformation, and are hence usually refered to as Fourier-transform-limited. They are required in the typical experiments of coherent optical spectroscopy, such as optical nutation, free induction decay, or photon echoes (Brewer, 1977). Here, the theoretical treatments generally adopt a semiclassical procedure, using a density matrix or Bloch formalism to describe the molecular system subject to a pulsed or continuous classical optical field, which generates a macroscopic sample polarization. In principle, a fully quantal description is possible if one represents the state of the field by the coherent or quasi-classical state vectors (Glauber, 1965 Freed and Villaeys, 1978). For our purpose, however. 

The Heisenberg uncertainty principle is captured by the fact there are many different choices for/(0,x). Equivalently, the quantum-classical correspondence is not unique this is why, for example, there are two (and even more) equivalent forms for the kinetic energy (Equations 1.47 and 1.48). All/(0,x) that are consistent with Equation 1.54 provide a suitable classical correspondence the most popular choice,y(0,T) = 1, corresponds to the Wigner quasi-probability distribution function. ... 

Equation (28) gives the (absolute) rate constants for site-to-site transitions in the quasi-classical approximation. Evaluation of the Ry requires integration of the distribution function p(r) over the volume of site i and crest surface fiy separating sites i and j. 

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